Two main facts are needed here:
1. The logarithm  , regardless of the base of the logarithm, exists for
, regardless of the base of the logarithm, exists for  .
.
2. The square root  exists for
 exists for  .
.
(in both cases we're assuming real-valued functions only)
By (2) we know that  exists if
 exists if  , or
, or  .
.
By (1), we know that  exists if
 exists if  , or
, or  . But as long as the square root exists, it will always be positive, so this condition will always be met.
. But as long as the square root exists, it will always be positive, so this condition will always be met.
Ultimately, then, we only require  , so the function has domain
, so the function has domain  .
.
To determine the range, we need to know that, in their respective domains,  and
 and  increase monotonically without bound. We also know that
 increase monotonically without bound. We also know that  at minimum, at which point the square root term vanishes, so the least value the function takes on is
 at minimum, at which point the square root term vanishes, so the least value the function takes on is  . Then its range would be
. Then its range would be  .
.